The case \(H^0(L)=0\)  follows as in the computation for \(S^1\times S^2\) , [1]}. The case \(H^0(L)\ne 0\)  follows from our description of the moduli spaces in the same way as in the computation for flat manifolds in [1]}. The generators of the Floer chain complex are in bijection with the positive eigenvalues of the perturbed Dirac operators of the critical points of the Morse function \(f:\mathbb {T}\rightarrow \mathbb {R}\) . Consider now two consecutive critical points \(p,q\) , respectively a maximum and a minimum of \(f\) ; recall that the minimum is one of the connections \(B_*\in \mathbb {T}\)  for which \(D_{B_*}\)  has kernel. Given two stable critical points lying over them with relative grading one, there is exactly one point in the moduli space of trajectories connecting them, so the map is given by
\({U^k\cdot }{\longrightarrow } {equation*}where k is the complex spectral flow of the one parameter family of perturbed Dirac operators (and the gradings are shifted so the map has degree -1). The description of Lemma \ref {pertdir} identifies the spectral flow with the kernel of the Dirac operator D_{B_*}, hence the description in terms of eigenvalues of \tilde{\varphi }^* (notice that there is spectral flow on exactly one of the two trajectories converging to B_*). Finally, orientations are readily determined because we work entirely with reducible moduli spaces.\) 
From this description it is straightforward to compute the corresponding Floer homology groups; the key observation is that when \(H^0(L)\ne 0\) , in degrees high enough the chain complex looks like
\(\begin{tikzcd}[column sep=tiny]\mathbb {Z}[dr]&&\mathbb {Z}^2 [dl][dr]&&\cdots &&&\mathbb {Z}^2 [dr] [dl]&&\mathbb {Z}[dl]\\& \mathbb {Z}^2&&\mathbb {Z}^2&\cdots &&\mathbb {Z}^2&&\mathbb {Z}^2\end{tikzcd}\) 
where all the maps are either the identity or the diagonal inclusion; this is exactly the Morse complex of a circle with a function with exactly \(2k\)  minima, so the homology is that the circle.
Let us discuss some concrete examples, see Figure REF .
<FIGURE>Example 3 
Suppose there is a single eigenvalue \(\lambda _1=e^{\pi i r_1}\)  with \(r_1>0\)  of multiplicity \(m_1\) . Then, taking into account the gradings, the chain complex is given by
-2m1 [dr][dl]
 2
and we compute
\(\widehat{\mathit {HM}}\) 

180
\(\widehat{\phantom{\mathit {HM}}}\) 
 HM
HM(M,sL)=1-2m1

and the action of \(\gamma \)  is surjective from the second tower onto the first.

Remark 8 In this example, and all the following ones, the computation for the conjugate spectrum (i.e. changing sign to all the \(r_i\) ) is obtained by reflecting the picture of the chain complex with respect to a vertical line.

Example 4 
Suppose there are two eigenvalues \(\lambda _i=e^{\pi i r_i}\)  of multiplicity \(m_i\)  with \(r_i>0\) . Then the chain complex with gradings is given by
-2(m1+m2) [dr]2-2m1[dl][dr][dl]
 2-2m12
and we compute
\(\widehat{\mathit {HM}}\) 

180
\(\widehat{\phantom{\mathit {HM}}}\) 
 HM
HM(M,sL)=1-2(m1+m2)

and the action of \(\gamma \)  is surjective from the second tower onto the first.

Remark 9 In general, for homology computations a sequence of consecutive eigenvalues \(\lambda _i=e^{\pi i r_i},\dots , \lambda _j=e^{\pi i r_j}\)  with \(r_i,\dots ,r_j\)  of the same sign is equivalent to a single eigenvalue of the same sign and multiplicity \(m_i+m_{i+1}+\dots + m_j\) .

Example 5 
Suppose there are two eigenvalues \(\lambda _i=e^{\pi i r_i}\)  of multiplicity \(m_i\)  with \(r_1<0<r_2\) . Then the Floer chain complex is
-2m2 [dr]2[dl][dr]-2m1[dl]
 22
which has homology
\(\widehat{\mathit {HM}}\) 

180
\(\widehat{\phantom{\mathit {HM}}}\) 
 HM
HM(M,sL)=1-2MZ[U]/Um

where \(M\)  and \(m\)  are respectively the maximum and the minimum of \(m_1,m_2\) ; here the top element of the last summand lies in degree \(-1\) .

Example 6 
Suppose there are two eigenvalues \(\lambda _i=e^{\pi i r_i}\)  of multiplicity \(m_i\)  with \(r_1>0>r_2\) . Then the chain complex is
2m2 [dr]2[dl][dr]2m1[dl]
 22m222m1
which has homology
\(\widehat{\mathit {HM}}\) 

180
\(\widehat{\phantom{\mathit {HM}}}\) 
 HM
HM(M,sL)=1-2MZ[U]/Um,

where the bottom elements of the two last summands lie in the same grading. Notice that after changing the orientation we obtain a manifold belonging to Example REF , and indeed (REF ) and (REF ) are obtained from each other via Poincaré duality.

Our description readily adapts to compute the Floer homology with respect to a non-trivial local system. For example, choosing a 1-cycle \(\eta \)  with \([\eta ]\ne 0\in H^1(M_{\varphi };\mathbb {R})\) , we can consider the completed Floer homology \(HM_{\bullet }(M_{\varphi },\mathfrak {s}_L;\Gamma _\eta )\)  with coefficients in the local system \(\Gamma _\eta \)  with fiber \(\mathbb {R}\) ; the key point is that in this case the analogue with local coefficients of the chain complex of the circle in Equation \((\ref {circchain})\)  has vanishing homology. We obtain:

in Examples REF  and REF , a cyclic module \(\mathbb {R}[U]/U^{m}\)  where \(m\)  is \(m_1\)  or \(m_1+m_2\)  respectively;

in Example REF , the rank two module \(\mathbb {R}[U]/U^{M}\oplus \mathbb {R}[U]/U^{m}\)  (the top elements have the same grading);

in Example REF , the rank two module \(\mathbb {R}[U]/U^{M}\oplus \mathbb {R}[U]/U^{m}\)  (the bottom elements have the same grading).

With these examples in mind, we are ready to prove the results from the Introduction.
Consider the bottom degrees in which the Floer chain complex looks like (REF ), and set them to be 0 and \(-1\)  respectively. Below this degrees, the Floer chain complex looks like a direct sum of chain complexes of the form
[column sep=tiny]
Z[dr]Z[dl][dr]Z[dr] [dl]Z[dl]
 ZZZZ
where all maps are isomorphisms, and the top row lies in even absolute \(\mathbb {Z}_2\) -grading; this has homology \(\mathbb {Z}\)  lying in the top grading represented by a signed sum of all generators in the top row. In particular the bottom of the tower in homology whose all elements are in odd grading has degree \(-1\) . The bottom of other the tower is then readily seen to be exactly the lowest grading among the towers corresponding to a maximum of the Morse function, which is exactly the quantity \(-2\Delta _L\)  in the statement of theorem .
Furthermore, the conjugation action induces an involution on the set of these chain subcomplexes; such actions has two kind of orbits:

pairs of complexes which are exchanged by the action; these are exactly those that do not involve critical points over the maxima at \(\pm 1\in \mathbb {T}\) ;

complexes which are sent into themselves by the action; in this case the central copy of \(\mathbb {Z}\)  lies over \(\pm 1\) .

Looking at the homology with local coefficents as in Theorem REF , each such subcomplex contributes a copy of \(\mathbb {R}\)  to the homology. Furthermore, each pair of `endpoints' of such complexes of contributes an extra dimension to \(H^0(L)\)  via a spectral flow consideration. This proves that the rank \(HM_{\bullet }(M_{\varphi },\mathfrak {s}_L;\Gamma _\eta )\)  is exactly \(h^0(L)\) . Finally, the statements about the ranks as \(U\) -modules follow from analogous arguments.

Computations
In this final section we discuss concrete computations of Floer homology that can be performed via our results by directly computing the action of the automorphism \(\varphi \)  on the space of holomorphic sections.
We begin with the case of an automorphism \(\varphi \)  of order 2, which corresponds to the case of hyperelliptic surfaces; such an automorphism leaves all spin structures invariant. By Lemma REF  (see also Example REF ), the lift \(\tilde{\varphi }\)  has order 4, and (after possibly taking the opposite lift) it acts on the space of harmonic spinors as multiplication by \(i\) . Via Example REF , Proposition REF  is then a direct consequence of the computation of the \(h^0(L)\)  for the corresponding linear systems, which can be found in [3]} and [4]}; the statement is the following.
Proposition 4.1 Consider a hyperelliptic Riemann surface of genus \(g\)  with hyperelliptic involution \(\varphi \) , and \([\eta ]\ne 0\) . Among the \(2^{2g}\)  theta characteristics \(L\) :

exactly \(\binom{2g+1}{g}\)  have \(h^0(L)=0\) ;

for any \(1\le w\le g\)  odd, there are exactly \(\binom{2g+2}{g-w}\)  theta characteristics \(L\)  with \(h^0(L)=(w+1)/2\) .


The proof involves an explicit description of \(2^{2g}\)  divisors corresponding to the square roots of the canonical bundle in terms of the \(2g+2\)  fixed points of the hyperelliptic involution.
In light of our results, for a general automorphisms of order \(\ge 3\)  the Floer homology groups will depend not only on the dimension of \({H}^0(L)\)  but also on the spectrum of the action of \(\tilde{\varphi }^*\) . While it is in general quite challenging to compute such data, we can use the Atiyah-Bott \(G\) -spin theorem [5]} to obtain useful information about it. We start by briefly recalling the geometric setup, which is more natural in the context of the spaces of harmonic spinors \(\mathcal {H}^{\pm }\) .
Given a finite order spin automorphism \(\varphi \)  (with lift \(\tilde{\varphi }\) ) of a \(2m\) -dimensional spin manifold, we obtain a natural action \(\tilde{\varphi }^*\)  on the spaces \(\mathcal {H}^{\pm }\)  of harmonic spinors. Its character is then defined to be
\(\chi (\tilde{\varphi }^*)=\mathrm {tr}(\tilde{\varphi }^*\vert _{\mathcal {H}^+})-\mathrm {tr}(\tilde{\varphi }^*\vert _{\mathcal {H}^-})\in \mathbb {C}.\) 
Of course \(\chi (\mathrm {id}^*)\)  is simply the index of \(D^+\) , which can be computed using the Atiyah-Singer index theorem. A generalization of this, known as the \(G\) -spin theorem [5]}, allows to compute \(\chi (\tilde{\varphi }^*)\)  in terms of the fixed locus of \(\varphi \)  and the action of \(d\varphi \)  on the normal bundle of the fixed locus. In general, such a formula is very complicated to use in concrete situations because the local contributions come with a sign (where the lift \(\tilde{\varphi }\)  comes into play) that is hard to determine. Luckily, in [7]} the authors worked out exactly the sign contribution in the case of a hyperbolic \(2m\) -manifold for automorphisms that lift to spin automorphisms of the same order. Via Lemma REF , we can use this to study a spin action of odd order on a compact Riemann surface. The result is the following, where we consider the unique hyperbolic metric inducing the given conformal structure (notice that uniqueness implies that the metric is \(\varphi \) -invariant).
Proposition 4.2 (Theorem \(8.1\)  of [7]}) 
Consider an automorphism \(\varphi \)  of odd order \(d\)  acting on a spin Riemann surface, and consider the lift \(\tilde{\varphi }\)  with the same order. Suppose that \(\varphi \)  has fixed points \(p_1,\dots ,p_m\) , and the action of \(d\varphi _{p_i}\)  on \(T_{p_i}\)  is given by rotation by \(\theta _i\) , where \(0<|\theta _i|<\pi \) . Then
\(\chi (\tilde{\varphi }^*)=i\sum _i\cos (\tilde{\theta }_i)\csc (\theta _i)=\frac{i}{2}\sum _i\csc (\tilde{\theta }_i),\) 
where we denote by \(\tilde{\theta }_i\)  the square root of \(\theta _i\)  for which \(d\cdot \tilde{\theta }_i\in 2\pi \mathbb {Z}\) .

Remark 10 The formula implies that \(\chi (\tilde{\varphi }^*)\in i\mathbb {R}\) ; this is consistent with the fact that on a surface the actions of \(\tilde{\varphi }^*\)  on \(\mathcal {H}^{\pm }\)  are dual to each other, cf. Remark REF .

Remark 11 Notice that the theorem does not assume that \(\Sigma /\varphi =\mathbb {P}^1\) . In general, given a \(\varphi \) -invariant spin structure it is already an interesting question to determine whether such spin structure is even or odd (cf. [9]} for the case in which the quotient is an elliptic curve).

We will now show how to use this theorem to compute the Floer homology for all automorphism of prime odd order; we begin by proving Proposition REF  and REF .
If the rotation angle at a fixed point is \(\theta =e^{\pm 2\pi i/3}\) , the corresponding half-angle is given by \(\tilde{\theta }=e^{\mp 2\pi i/3}\) ; as \(\csc (\pm 2\pi /3)=\pm 2\sqrt{3}/3\) , the character of \(\tilde{\varphi }^*\)  is
\(\chi (\tilde{\varphi }^*)=\frac{i\sqrt{3}}{3}(n_--n_+).\) 
By Lemma REF , \(\varphi \)  acts on \(H^0(L)\)  as multiplication by either \(e^{\pm 2\pi i/3}\) , in which case the character of \(\varphi \)  is
\(\pm (e^{ 2\pi i/3}-e^{- 2\pi i/3})\cdot h^0(L)=\pm i\sqrt{3}\cdot h^0(L).\) 
So \(H^0(L)\)  has dimension \(|n_+-n_-|/3\) , and the conclusion follows from the computation in Example REF .
In this case the character \(\chi (\tilde{\varphi }^*)\)  will be an integral combination of
\(e^{2\pi i/5}-e^{-2\pi i/5}=2i\sin (\frac{2\pi }{5})&=i\sqrt{\frac{5+\sqrt{5}}{2}}\\e^{4\pi i/5}-e^{-4\pi i/5}=2i\sin (\frac{4\pi }{5})&=i\sqrt{\frac{5-\sqrt{5}}{2}}.\) 

As these are linearly independent over \(\mathbb {Q}\) , and the spectrum of \(\tilde{\varphi }^*\)  contains only one of in each pair \(e^{\pm 2\pi i/5},e^{\pm 4\pi i/5}\)  by Lemma REF , the spectrum is determined uniquely by \(\chi (\tilde{\varphi }^*)\) . We compute
\(\csc (\widetilde{2\pi /5})/2&=-\frac{1}{5}\cdot 2\sin (2\pi /5)-\frac{2}{5}\cdot 2\sin (4\pi /5)\\\csc (\widetilde{4\pi /5})/2&=\frac{2}{5}\cdot 2\sin (2\pi /5)-\frac{1}{5}\cdot 2\sin (4\pi /5),\) 

where \(\widetilde{2\pi /5}=6\pi /5\)  and \(\widetilde{4\pi /5}=2\pi /5\) . Hence, if the automorphism has \(p\)  fixed points of rotation \(2\pi /5\)  and \(q\)  fixed points of rotation \(4\pi /5\)  (where we count with signs), then the character is given by
\(\chi (\tilde{\varphi }^*)=i\left(\frac{-p+2q}{5}\cdot 2\sin (\frac{2\pi }{5})+\frac{-2p-q}{5}\cdot 2\sin (\frac{4\pi }{5}) \right).\) 
We can construct examples in which the action of \(\tilde{\varphi }^*\)  on \(\mathcal {H}^+\)  has eigenvalues \(e^{2\pi i/5}\)  and \(e^{4\pi i/5}\)  with the desired multiplicities \(m,n\in \mathbb {Z}\)  (where negative multiplicity corresponds to the conjugate eigenvalue) by taking
\(p=-m-2n\quad \text{and}\quad q=2m-n.\) 
We can write down such automorphism explictly as the natural \(\mathbb {Z}/5\) -action on a curve of the form \(y^5=f(x)\)  for a suitable polynomial \(f\) , following [10]}. Here, near a root of \(f\)  of multiplicity \(a\)  the local model of the singularity of the curve is \(z^5=w^a\) , and the local rotation is given by \(b\cdot 2\pi /5\)  where \(b\)  is the inverse of \(a\)  modulo 5. The conclusion of the Proposition follows from the computations in Example REF  and REF .
To conclude, we discuss the proof of Theorem REF , whose proof is a generalization of the method we used in the examples above. The main observation is the following.
Lemma 4.3 Let \(p\)  an odd prime, and let \(\zeta =e^{2\pi i/p}\) . Then the numbers \(\zeta ^i-\zeta ^{-i}\)  for \(i=1,\dots , (p-1)/2\)  are linearly independent over \(\mathbb {Q}\) .

Suppose there is a non-trivial relation with rational coefficients \(a_1,\dots , a_{p-1/2}\) . Then
\(a_1(\zeta -\zeta ^{-1})+a_2(\zeta ^2-\zeta ^{-2})+\dots +a_{(p-1)/2}(\zeta ^{(p-1)/2}-\zeta ^{-(p-1)/2})=0.\) 
Because there is no constant term, this reduces to a polynomial of degree \(<p-1\)  satisfied by \(\zeta \) . This is a contradiction because \(\zeta \)  has minimal polynomial \(1+\zeta +\zeta ^2+\dots +\zeta ^{p-1}\)  over \(\mathbb {Q}\) .
Given an automorphism of odd prime order \(p\) , we can use the \(G\) -spin theorem to compute \(\chi (\tilde{\varphi }^*)\)  in terms of ramification data. Now, \(\chi (\tilde{\varphi }^*)\)  is an integral linear combination of \(\zeta ^i-\zeta ^{-i}\)  for \(i=1,\dots (p-1)/2\) , and by the previous lemma the coefficients are uniquely determined. Furthermore, no conjugate pairs \(\zeta ^i,\zeta ^{-i}\)  appear in the spectrum, so we obtain that \(\chi (\tilde{\varphi }^*)\)  determines completely the spectrum of the action of \(\tilde{\varphi }^*\)  on \(H^0(L)\) ; we can then use this to determine the Floer chain complex and its homology as in Section .

